The present invention relates generally to estimating surface water runoff, and, in particular, to deriving an instantaneous unit hydrograph for a watershed.
Rainfall is partially absorbed into and partially shed by the surface on which it falls. The proportion that is shed will depend on how long and how much it rains at one time, on the type of surfaces on which it falls, the slope of the surface, on the condition of the surface at the particular time it rains (already saturated surfaces absorb more, for example) and on other factors. The water that is shed may have to be managed in some cases rather than allow for it to simply flow into a down-slope stream, river, lake or sea. Management of runoff requires physical structures that control, redirect, or confine the surface water and that protect adjacent areas.
In order to manage surface water runoff, an estimate of the amount of runoff is useful. Structures that manage the runoff will be sized to receive the estimated volume of runoff. The larger the estimated runoff, the larger the structures need to be built in order to cope with it. Also, the size of the structures is typically increased to allow for uncertainty in the estimated runoff.
If surface water could be more accurately estimated, the costs of structures built to manage it can be lowered because they could be built with less margin for uncertainty. Furthermore, the costs associated with the consequences of a structure being under-designed are also reduced. For example, with a more accurate estimate, the structure might be designed to be smaller and therefore require less space and fewer construction materials. On the other hand, a more accurately designed structure may prevent the washing out of roads and the attendant repairs and inconvenience of detours while those repairs are made.
In order to estimate runoff, hydrogeologists attempt to determine how fast rainfall excess occurring uniformly over a particular watershed will reach its outlet. The speed will depend heavily on the nature and number of flow paths, both overland and channel flow paths, that the excess rainfall follows.
This determination can be done by placing gauges at various locations in a watershed to measure rainfall and runoff. However, this type of study is not always practical because of the time and resources involved. In many cases, geohydrologists must rely on estimates made by a mathematical analysis. As a practical matter, this analysis cannot be precise but must employ certain simplifying assumptions.
Estimating surface water runoff involves making a number of assumptions about the weather and combining these assumptions with information about the area on which rain falls. These two components can be viewed separately by using a diagram called a unit hydrograph. A unit hydrograph shows what volume of water as a function of time reaches a drain, or xe2x80x9ccatchment,xe2x80x9d in a watershed following one unit of rainfall. A watershed is a topographically defined region where all surface water tends to flow to a single drain point. For example, a watershed may be a valley where all of the surface water drains to a stream in its lowest point and thence to some other area. This type of graph says nothing about the anticipated weather but is solely directed to what happens to rainfall if it occurs. An xe2x80x9cinstantaneousxe2x80x9d unit hydrograph assumes that the unit of rainfall occurs instantaneously.
To simplify matters conceptually, hydrogeologists assume that a unit of rainfall falls uniformly over the whole watershed. The instantaneous unit hydrograph may then be determined by taking the time derivative of the volume of flow at the outlet that results from the unit of rainfall that has fallen in an instant. Another way of stating the problem is: What is the probability that a drop of rainfall excess has reached the watershed outlet at some time t? The answer is given by the equation:       V    ⁢          (      t      )        =            ∫      0      t        ⁢                  q        ⁢                  (          t          )                    ⁢              xe2x80x83            ⁢              ⅆ        t            
where q(t) is given by the formula       q    ⁢          (      t      )        =            IUH      ⁢              (        t        )              =                  ⅆ                  V          ⁢                      (            t            )                                      ⅆ        t            
where V(t) is the total volume of rainfall excess at the outlet up to time t and q(t) is the discharge hydrograph at some time t.
Unit hydrographs were first developed in the early 1930""s by L. K. Sherman as a way to transform rainfall into runoff. Sherman based his model for hydrographs on observed rainfall in a watershed and the corresponding outflow.
Unit hydrographs are often made in the same way today, that is, by making measurements over a period of time. Records of rainfall can be correlated to surface water outflow at the drain from the basin. However, it is not always possible to make actual measurements of every basin. When measurements are not feasible, unit hydrographs must be derived indirectly or xe2x80x9csynthesizedxe2x80x9d about a watershed using other information. Synthesized unit hydrographs are developed for ungauged watersheds using statistical parameter prediction equations that relate unit hydrographs from gauged watersheds.
In order to perform this analysis, some additional terms are needed. The word xe2x80x9cstatexe2x80x9d refers to the order of the overland flow region or the channel in which the drop is located at time t. The number of the state is determined by the number of linear reservoirs used to define the overland segment of flow. This number can be varied so that the shape of the unit hydrograph can be better approximated. All drops of water eventually pass into the highest numbered, or xe2x80x9ctrapping,xe2x80x9d state N where xcexa9 is the number of states used to represent overland and channel flow for the entire basin and, thus, N=xcexa9+1. The term xe2x80x9ctransitionxe2x80x9d means that the state of the drop has changed.
A major improvement in synthesizing unit hydrographs occurred when Horton in 1945 introduced the use of order numbers and ratios for flow channels. His method was further refined by Strahler in 1957. According to this method, channels that originate at a source are first order streams. When two streams of order i join, a stream of order i+1 is created. Finally, when two streams of different order join, the stream immediately downstream of where they join is assigned the higher of the orders of the two joining streams. This will be referred to herein as the Horton-Strahler method.
Horton proposed that for a given basin with its network of channels, the number of streams of successive orders and the mean lengths of streams of successive orders can be approximated by simple geometric progressions. The mean length Li of a stream of order i is defined by       L    i    =            1      N        ⁢                                        xe2x80x83                    ∑                          j          =          1                          N          i                    ⁢              xe2x80x83            ⁢              L        ji            
where Lij, j=1, 2, . . . Ni, i=1, 2, . . . , xcexa9, represents the length of the jth stream of order i.
Horton established three ratios, RB, RL and RA The first, RB, the bifurcation ratio, is the Horton xe2x80x9claw of stream numbersxe2x80x9d:       R    B    ≅            N              i        -        1                    N      i      
The first Horton ratio is typically in the range of 3 and 5 for natural areas.
The second ratio, RL, is the stream length ratio for the Horton xe2x80x9claw of stream lengthsxe2x80x9d:       R    L    ≅            L      i              L              i        -        1            
The RL ratio for natural areas is typically between 1.5 and 3.5.
A third ratio, RA, proposed by Schuum in 1956 and called the Horton xe2x80x9carea ratioxe2x80x9d, is the drainage area ratio:       R    A    ≅            A      i              A              i        -        1            
RA is found in a manner similar to that of RB and RL. This third Horton ratio is typically between 3 and 6 for natural areas.
In this equation, the area Ai is the mean area of the basin region of order i. Specifically,       A    i    =            1              N        i              ⁢          xe2x80x83        ⁢          ∑              xe2x80x83            ⁢              A        ji            
for i=1,2, . . . , xcexa9. Aij refers to the total area that drains eventually into the jth stream of order i and not just the area of the surface region that drains directly into the jth stream of order i. Consequently, Ai greater than Aixe2x88x921.
There are several methods known for developing synthetic unit hydrographs, some of which employ the Horton ratios. However, the movement of water through a basin is a very complex process. Hydrologic systems are not linear, as assumed by the simpler models. The characteristics that explain the non-linearities in watershed response need to be identified and the form of the mathematical functions used to represent them chosen, or otherwise the effective use of the models would continue to be limited to watersheds similar to those from which the models were developed.
The involvement of watershed geomorphology has proved to be a significant advance in unit hydrograph modeling. The first geomorphologic instantaneous unit hydrograph was developed by Rodriguez-Iturbe and Valdes in 1979 (xe2x80x9cThe Geomorphologic Structure of Hydrologic Response,xe2x80x9d Water Resource Research, Vol. 15, No. 6, December 1979, p. 1409). It expressed the unit hydrograph as a function of the Horton Order Ratios following the Strahler stream-ordering system developed in 1957, an internal scaling parameter, and a mean velocity streamflow. It classified streams in a network of linear reservoirs. Then, it modeled the movement of water in the network with transition probabilities. Travel time was conceptualized as a holding or waiting time, and evaluated as the mean travel time for each order stream. The watershed geomorphology determined the basic instantaneous unit hydrograph shape. Constant velocity was assumed; overland flow was neglected.
Others, such as Lee and Yen (xe2x80x9cGeomorphology and Kinematic-Wave-Based Hydrograph Derivation,xe2x80x9d Journal of Hydraulic Engineering, January 1997, p. 73) have considered overland flow and variable flow in unit hydrograph modeling by incorporating topographic maps and remote sensing to provide information about overland surfaces and gradients. However, all of these studies were directed at natural basins, leaving urban areas essentially unstudied. In particular, the Horton ratios seem to work well for natural areas but are completely unsatisfactory for urban areas.
Where accurate unit hydrographs are needed most, namely, urban areas, they are the least available. Thus, there remains a need for a way to accurately synthesize unit hydrographs for urban areas.
According to its major aspects and briefly recited, the present invention is a method of synthesizing geomorphological instantaneous unit hydrographs that applies to urban areas as well as natural areas. The method can account for overland flow, which is of particular importance in modeling urban areas, and for variations in velocity of the flow. Most importantly, in connection with urban areas, the input will result in a more accurate unit hydrograph than that obtained heretofore, and the input is readily obtainable from commonly available data and site inspection.
In the embodiment of the present method suitable for urban areas, the initial state matrix can be populated with area ratios, the transition matrix can be populated with ratios of the numbers of streams of each order, and overland travel time can be calculated directly from the input of velocities of flow and the lengths of the flow paths.
In an alternative embodiment of the present invention, if the Horton ratios for the basin of interest are within normal ranges, they can be used in the derivation of the geologic unit hydrograph. If, however, they are outside the normal ranges, the actual characteristics of the basin should be used instead. Typically, urban watersheds have Horton ratios that are outside the normal ranges.
For urban watersheds, map data are used to determine total areas draining directly into each order stream and the total area of the basin. If the map data include topographic data, they can also be used for determining gradients and thus flow velocities. The present program uses this information to determine the elements in the initial probabilities matrix and mean waiting times.
Transition probabilities are determined by the ratios of the numbers of streams of a particular order draining directly into streams of another order to the total number of streams of that particular order. The product of the initial probabilities and the transition probabilities is the state probability matrix. The derivative of the state probability hydrograph at the outlet with respect to time yields the instantaneous unit hydrograph.
Although the basic analysis is similar to the Horton-Strahler method as modified by Rodriguez-Iturbe/Valdez, the input for urban areas (or those areas where Horton Ratios are not within normal ranges), the use of variable velocities, and the ability to include in a practical way runoff from overland flow in the determination of the hydrograph are the significant features of the present invention.
With a more accurate unit hydrograph and historical weather data, the user can specify the requirements for surface water management structures. These structuresxe2x80x94culverts, reservoirs, channels, leveesxe2x80x94will not need to be designed with as much conservatism to account for uncertainty in runoff volume and are less likely to be under-designed because of faulty analysis. Therefore the cost in terms of resources in constructing and repairing these structures and their surroundings is likely to be lower than in the case of prior art analyses.
Other features and their advantages will be apparent to those skilled in the art of unit hydrograph derivation from a careful reading of the Detailed Description of Preferred Embodiments, accompanied by the drawings.